Optimal Mean-Variance Portfolio Selection

  • Jingsi Xu

Student thesis: Phd


This thesis studies the dynamic optimality introduced by Pedersen and Peskir [43] in the mean-variance portfolio selection problem from the dynamic programming perspective. For a self-financing portfolio, the investor aims to determine the maximal value function define by: V(t,x)=sup_u(E(X^u_T)-cVar(X^u_T)) The quadratic nonlinearity introduced by the variance term can be handled by the method of Lagrange multipliers, and the application of the HJB equation enables us to obtain the optimal solution. In this thesis, we introduce various different market settings, including (a) prohibition of short-selling, (b) margin requirements, (c) that the stock that is driven by the constant elasticity of variance model, and (d) partial information. For these problems, we investigate (i) time-inconsistent strategies (static optimality) and (ii) the time-consistent strategies (dynamic optimality), and we compare their performance. Alongside solving the original optimal problem, we also consider the other two constrained cases where we condition the size of the expectation/variance of the terminal wealth. In Chapter 2, we consider portfolio selection under a no short-selling constraint, in which a change-of variable formula from [45] is used to replace the viscosity solution to overcome the non-smoothness of the value function. Under the no short-selling constraint, both static and dynamic optimalities naturally prevent bankruptcy. However, static optimality suggests that the investor should hold all of his wealth in the riskless bond if his wealth is large enough, while dynamic optimality encourages the investor to keep holding the risky asset for a higher return. Inspired by the method applied in Chapter 2, we further consider the margin requirement for short-selling in Chapter 3. The conclusion analyses the impact of the change of margin rate on the performance of both static and dynamic optimalities and verifies that some properties of those two optimalities described in [43] are still valid in this case. In Chapter 4, we study the case where the stock price follows the constant elasticity of variance (CEV) model. The CEV model can be taken as a natural extension of geometric Brownian motion, and it has advantages such as explaining the implied volatility smile. We derive static and dynamic optimalities for both the unconstrained problem and the constrained problems. By choosing a proper value for the elasticity parameter, we can easily extend our conclusion to cases where the risky asset follows different processes such as geometric Brownian motion and the Ornstein-Uhlenbeck process. Hence, the model we set in Chapter 4 can be seen as a general solution that covers the work of [43]. Besides, the conclusion in Chapter 4 is also valid when there exist arbitrage opportunities for the stock. In Chapter 5, we consider portfolio selection under partial information, since the investor normally can only access limited information in a real financial market. The biggest difficulty is that the filtering and optimisation aspects of the problem are hard to separate, which can be handled by using the separation principle studied in [56]. Under partial information, we obtain both time-consistent and time-inconsistent solutions.
Date of Award1 Aug 2021
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorGoran Peskir (Supervisor) & Ronnie Loeffen (Supervisor)


  • The Hamilton-Jacobi-Bellman equation
  • The change of variable formula with local time on curves
  • Constant elasticity of variance model
  • Dynamic programming
  • Mean-variance portfolio selection
  • Dynamic optimality
  • Static optimality
  • Constrained nonlinear optimal control

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